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Deluxe Pythagorean Theorem

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by HeyArnold » Sun Aug 21, 2011 12:13 pm
Hello,

In an explanation to an OG question, Manhattan companion guide refers to a "deluxe pythagorean Theorem" as a means to determine the distance between two corners of a rectangular cube.

https://www.beatthegmat.com/deluxe-pytha ... 46196.html

In another Manhattan Problem, this time involving a square cube, this formula does not work.

https://www.beatthegmat.com/diagonals-of ... 82794.html

In which instances can this formula be applied?

Thanks!
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by GMATGuruNY » Mon Aug 22, 2011 3:10 am
HeyArnold wrote:Hello,

In an explanation to an OG question, Manhattan companion guide refers to a "deluxe pythagorean Theorem" as a means to determine the distance between two corners of a rectangular cube.

https://www.beatthegmat.com/deluxe-pytha ... 46196.html

In another Manhattan Problem, this time involving a square cube, this formula does not work.

https://www.beatthegmat.com/diagonals-of ... 82794.html

In which instances can this formula be applied?

Thanks!
The formula applies to both problems.

The main diagonal of a rectangular solid is the longest line segment that can be drawn through the solid.
The determine the length of the main diagonal, use the Super-Pythagorean theorem.
If L=length, W=width, H=height, and D=diagonal, then:

L^2 + W^2 + H^2 = D^2

or

D = √(L^2 + W^2 + H^2).

In a cube, L=W=H.
If e = length of each edge, then:

e^2 + e^2 + e^2 = D^2

or

D = √(3e^2).

I used this last version of the formula in my solution to the problem about the cube:

https://www.beatthegmat.com/diagonals-of ... 82794.html
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